divided into inches, tenths, and hundredths, for measuring the length of the column at all times, by observing which division of the scale the top of the quicksilver is opposite to; as it ascends and descends within these limits, according to the state of the atmosphere. So the weight of the quicksilver in the tube, above that in the basin, is at all times equal to the weight or pressure of the column of atmosphere above it, and of the same base with the tube; and hence the weight of it may at all times be computed; being nearly at the rate of half a pound avoirdupois for every inch of quicksilver in the tube, on every square inch of base; or more exactly, it is of a pound on the square inch, for every inch in the altitude of the quicksilver: for the cubic inch of quicksilver weighs just lb., or nearly a pound, in the mean temperature of 55° of heat. And consequently, when the barometer stands at 30 inches, or 2 feet high, which is the medium or standard height, the whole pressure of the 59 120 31 28 atmosphere is equal to 14 pounds, on every square inch of the base. And so in proportion for other heights. OF THE THERMOMETER. 55. THE THERMOMETER is an instrument for measuring the temperature of the air, as to heat and cold. It is found by experience, that all bodies expand by heat, and contract by cold and hence the degrees of expansion become the measure of the degrees of heat. Fluids are more convenient for this purpose, than solids: and quicksilver is now most commonly used for it. A very fine glass tube, having a pretty large hollow ball at the bottom, is filled about half way up with quicksilver: the whole being then heated very hot till the quicksilver rise quite to the top, the top is then hermetically sealed, so as perfectly to exclude all communication with the outward air. Then, in cooling, the quicksilver contracts, and consequently its surface descends in the tube, till it come to a certain point, correspondent to the temperature or heat of the air. And when the weather becomes warmer, the quicksilver expands, and its surface rises in the tube; and again contracts and descends when the weather becomes cooler. So that, by placing a scale of any divisions against the side of the tube, it will show the degrees of heat, by the expansion and contraction of the quicksilver in the tube; observing at what division of the scale the top of the quicksilver stands. And the method of preparing the scale, as used in England, is thus:-Bring the thermometer into a temperature of just freezing, by immersing the ball in water just freezing, or in ice just thawing, and mark the scale where the mer- This division of the scale, is commonly called Fahrenheit's. According to this division, 55 is at the mean temperature of the air in this country; and it is in this temperature, and in an atmosphere which sustains a column of 30 inches of quicksilver in the barometer, that all measures and specific gravities are taken, unless when otherwise mentioned; and in this temperature and pressure, the relative weights, or specific gravities, of air, water, and quicksilver, are as 12 for air, 1000 for water, and 13600 for mercury; and these also are the weights of a cubic foot of each, in avoirdupois ounces, in that state of the barometer and thermometer. 19 100 25 For other states of the thermometer, each of these bodies expands or contracts, according to the following rate, with each degree of heat; viz. OF THE MEASUREMENT OF ALTITUDES BY THE BAROMETER AND THERMOMETER. 56. FROM the principles laid down in the Scholium to prop. 17, concern. ing the measuring of altitudes by the barometer, and the foregoing descriptions of the barometer and thermometer, we may now collect together the precepts for the practice of such measurements, which are as follow: First, Observe the height of the barometer at the bottom of any height, or depth, intended to be measured; with the temperature of the quicksilver by means of a thermometer attached to the barometer, and also the temperature of the air in the shade by a detached thermometer. Second, Let the same thing be done also at the top of the said height or depth, and at the same time, or as near the same time as may be. And let those altitudes of barometer be reduced to the same temperature, if it be thought necessary, by correcting either the one or the other, that is, augment the height of the mercury in the colder temperature, or diminish that in the warmer, by its part for every degree of difference of the two. Third, Take the difference of the common logarithms of the two heights of the barometer, corrected as above if necessary, cutting off three figures next the right hand for decimals, the rest being fathoms in whole numbers. Fourth, Correct the number last found for the difference of temperature of the air, as follows:-Take half the sum of the two temperatures, for the mean one; and for every degree which this differs from the temperature 31o, take so many times the part of the fathoms above found, and add them if the mean temperature be above 31o, but subtract them if the mean temperature be below 31°; and the sum or difference will be the true altitude in fathoms; or, being multiplied by 6, it will be the altitude in feet. EXAMPLE 1.-Let the state of the barometers and thermometers be as follows; to find the altitude, viz. EXAMPLE 2.-To find the altitude, when the state of the barometers and thermometers are as follows, viz. OF THE RESISTANCE OF FLUIDS, WITH THEIR FORCES AND ACTION ON BODIES. PROP. L. 57. If any body move through a fluid at rest, or the fluid move against the body at rest; the force or resistance of the fluid against the body, will be as the square of the velocity and the density of the fluid. That is, Rx dv2. FOR, the force or resistance is as the quantity of matter or particles struck, and the velocity with which they are struck. But the quantity or number of particles struck, in any time, are as the velocity and the density of the fluid. Therefore the resistance or force of the fluid, is as the density and square of the velocity. Corol. 1. The resistance to any plane, is also more or less, as the plane is greater or less; and therefore the resistance on any plane, is as the area of the plane a, the density of the medium, and the square of the velocity. That is, Rx adv. Cordl. 2. If the motion be not perpendicular, but oblique to the plane, or to the face of the body; then the resistance, in the direction of motion, will be diminished in the triplicate ratio of radius to the sine of the angle of inclination of the plane to the direction of motion, or as the cube of radius to the cube of the sine of that angle. So that Rx adv's3, putting 1 = radius, and s = sine of the angle of inclination CAB. For, if AB be the plane, AC the direction of motion, ▲ and BC perpendicular to AC; then no more particles meet the plane than what meet the perpendicular BC, and therefore their number is diminished as AB to BC, or as I to s. But the force of each particle, striking the plane obliquely in the direction CA, is also dimin B ished as AB to BC, or as I to s; therefore the resistance, which is perpendicular to the face of the plane, by art. 8 is as 12 to s2. But again, this resistance in the direction perpendicular to the face of the planes, is to that in the direction AC, by art. 8 as AB to BC, or as 1 to s. Consequently, on all these accounts, the resistance to the plane when moving perpendicular to its face, is to that when moving obliquely, as l3 to s3, or 1 to s3. That is the resistance in the direction of the motion, is diminished, as I to s3, or in the triplicate ratio of radius to the sine of inclination. PROP. LI. 58. The real resistance to a plane, by a fluid acting in a direction perpendicular to its face, is equal to the weight of a column of the fluid, whose base is the plane, and altitude equal to that which is due to the velocity of the motion, or through which a heavy body must fall to acquire that velocity. THE resistance to the plane moving through a fluid, is the same as the force of the fluid in motion with the same velocity, on the plane at rest. But the force of the fluid in motion, is equal to the weight or pressure which generates that motion; and this is equal to the weight or pressure of a column of the fluid, whose base is the area of the plane, and its altitude that which is due to the velocity. Corol. 1. If a denote the area of the plane, v the velocity, n the density or specific gravity of the fluid, and49= 16 feet, or 193 inches. Then, the altitude due to the velocity v being ,the whole resistance, or motive force R, v2 29 Corol. 2. If the direction of motion be not perpendicular to the face of the plane, but oblique to it, in an angle whose sine is s. Then the resistance Corol. 3. Also, if w denote the weight of the body, whose plane face R w a is resisted by the absolute force R; then the retarding force f, or will be anv2ss 2gw Corol. 4. And if the body be a cylinder, whose face or end is a, and radius r moving in the direction of its axis; because then s =1, and a = pr3, Corol. 5. This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direction of motion. But were its face an elliptic section, or a conical surface, or any other figure every where equally inclined to the axis, or direction of motion, the sine of inclination being s: then, the number of particles of the fluid striking the face being still the same, but the force of each, opposed to the direction of motion, diminished in the duplicate ratio of radius to the sine of inclination, the resistpnr*v* ing force R will be 29 PROP. LII. 59. The resistance to a sphere moving through a fluid, is but half the resist ance to its great circle, or to the end of a cylinder of the same diameter, moving with the same velocity. LET AFEB be half the sphere, moving in the direction CEG. Describe the paraboloid AIEKB on the same base. Let any particle of the medium meet the semi-circle in F, to which draw the tangent FG, the radius FC, and the ordinate FIH. Then the force of any particle on the surface at F, is to its force on the base at H, as the square of the sine of the angle G, or its equal the angle FCH, to the square of radius, that is, as HF" to CF. Therefore the force of all the particles, or the whole fluid, on the whole surface, is to its force on the circle of the base, A H F as all the HF to as many times CF. But CF" is = CA' = AC. CB, and HF' = AH. HB by the nature of the circle; also, AH. HB: AC. CB :: HI: CE by the nature of the parabola; consequently the force on the spherical surface, is to the force on its circular base, as all the HI's to as many CE's, that is, as the content of the paraboloid to the content of its circumscribed cylinder, as 1 to 2. being the Corol. Hence, the resistance to the sphere is R = half of that of a cylinder of the same diameter. For example, a 9lb iron ball, whose diameter is 4 inches, when moving through the air with a velocity of 1600 feet per second, would meet a resistance which is equal to a weight of 132 lbs., independent of the pressure of the atmosphere, for want of the counterpoise behind the ball. FINIS. BALNE BROTHERS, PRINTERS, GRACECHURCH STREET. |